Problem: If x and y are positive integers such that $xy = 100,$ what is the positive difference between the maximum and minimum possible values of $x + y?$
Explanation: We know that x and y are both positive integers, and that $xy=100. \ 100$ is small enough that we can reasonably find all the possible pairs of numbers: $100=2\cdot2\cdot5\cdot5$, so we could have $(1,100); (2, 50); (4,25); (5,20); (10; 10)$ or the reverse of any given pair for $(x,y).$ The possible sums are $1+100=101, \ 2+50=52, \ 4+25=29,$ and $10+10=20.$ The largest sum is $101$ and the smallest is $20$, with difference $101-20=\boxed{81.}$